Sec. 2.7 THE COEFFICIENT OF VARIATION 41 



In other words, the standard deviation of the yields in bushels per 

 acre is 16% times that of the same yields expressed as pounds per 

 1/1000 acre. Hence, even though o- is an excellent and widely used 

 measure of the variability exhibited by a group of numerical meas- 

 urements, its size does depend directly upon the units of measure 

 involved, and also upon the level of magnitude of those measurements. 

 To illustrate the point regarding the level of magnitude of measure- 

 ment, suppose that one were interested in knowing if the weights of 

 thirty-year-old males in Manhattan, Kansas, were more (or less) 

 variable than the weights of twelve-year-old boys in that city. Sup- 

 pose also that the average weight of the men is known to be approxi- 

 mately twice that of the youths. The analysis just presented shows 

 that if the boys' weights were each to be doubled so they would be 

 on a level comparable to that of the men, their standard deviation 

 automatically would be doubled too. It does not seem reasonable 

 that doubling all of the X's in a set of measurements should change 

 their fundamental variability relative to another set of measure- 

 ments; hence there is need for a measure of variability which would 

 not be so affected. The coefficient of variation is that sort of measure 

 of relative variability. 



It is easy to see that the mean of kX is k times the mean of X be- 

 cause fi kx = $(kX)/N = k%{X)/N = kfi x . Therefore, the ratio of 

 the standard deviation to the arithmetic mean will be a measure of 

 relative variability in a useful sense because 



okx _ ktrx _ vx 



MftZ kfJix MV 



regardless of the size of k{^ 0) . It is customary to express this ratio 

 of the standard deviation to the arithmetic mean as a percentage, and 

 to define the coefficient of variation (CV) by 



(2.71) CV = 100a/ M . 



To illustrate formula 2.71 from previously discussed data, the stu- 

 dent can verify that, for ACE scores, CV = 27.3 ; for the birth weights 

 of guinea pigs, CV = 25.2; and for problem 5, section 2.5, CV = 18.9, 

 each as a per cent. A person acquainted with ACE scores might then 

 observe that the scores at Kansas State College during 1947 were 

 relatively more variable than the national scores, which (it is sup- 

 posed for illustration) had CV = 20 per cent. Concerning the birth 

 weights, we might learn that some other group of these animals has 

 a standard deviation of only 15 grams, and hastily (and erroneously) 



